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Mirrors > Home > MPE Home > Th. List > 19.21h | Structured version Visualization version GIF version |
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See also 19.21 2170 and 19.21v 1915. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
Ref | Expression |
---|---|
19.21h.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
19.21h | ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.21h.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | nf5i 2115 | . 2 ⊢ Ⅎ𝑥𝜑 |
3 | 2 | 19.21 2170 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-10 2110 ax-12 2139 |
This theorem depends on definitions: df-bi 208 df-ex 1760 df-nf 1764 |
This theorem is referenced by: hbim1 2269 |
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